Allocating Jobs with Periodic Demand Variations

Abstract

In the context of service hosting in large-scale datacenters, we consider the problem faced by a provider for allocating services to machines. Based on an analysis of a public Google trace corresponding to the use of a production cluster over a long period, we propose a model where long-running services experience demand variations with a periodic (daily) pattern and we prove that services following this model acknowledge for most of the overall CPU demand. This leads to an allocation problem where the classical Bin-Packing issue is augmented with the possibility to co-locate jobs whose peaks occur at different times of the day, which is bound to be more efficient than the usual approach that consist in over-provisioning for the maximum demand. In this paper, we provide a mathematical framework to analyze the packing of services exhibiting daily patterns and whose peaks occur at different times. We propose a sophisticated SOCP (Second Order Cone Program) formulation for this problem and we analyze how this modified packing constraint changes the behavior of standard packing heuristics (such as Best-Fit or First-Fit Decreasing). We show that taking periodicity of demand into account allows for a substantial improvement on machine utilization in the context of large-scale, state-of-the-art production datacenters.

Column Generation Integer Programming for Allocating Jobs with Periodic Demand Variations

“…Smaller or short-lived jobs, which are much more numerous but represent a smaller part of the resource usage, can be handled with usual greedy allocation schemes. This paper is a followup to [4], which analyzes the performance of several standard packing heuristics in the context of packing jobs with periodic demand variation. In this paper, we propose the use of the Dantzig-Wolfe decomposition [17] to solve very efficiently the corresponding packing problem.…”

“…Based on this analysis, and following [4], we consider a packing problem for those long running jobs, which account for a large portion of the workload.…”

“…Due to the nature of this constraint, this formulation can be expressed as a Second Order Cone Program (SOCP) [12], and can thus benefit from efficient general purpose solvers [12] for convex optimization. However, as noticed in [4], on realsize instances with thousands of machines, this formulation can not be solved in reasonable time with integer and boolean values. Relaxing the problem by allowing rational variables makes it possible to obtain a lower bound on the necessary number of resources in reasonable time.…”

“…In this Section, we present some heuristics introduced in [4], adapted from classical efficient greedy Bin-Packing algorithms to the case of tasks exhibiting daily patterns. In the following, we denote by L(M k , T j,l ) the peak load on machine M k after adding one task T j,l of job J j to M k .…”

“…Previous work on the subject [4] argues that many services representing most of the overall CPU demand exhibit daily patterns and their demand can be modeled as a set of sinusoids, each comprising a constant component, an amplitude and a phase. This premise gives rise to a model for jobs with time-varying resource demands and to the associated packing problem.…”